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Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
x0
∈
omega
.
Apply unknownprop_513c07ab8c4a322509acfbace4471c9db9e0c1f3fed5cccfa0919534f9c289bd with
x0
,
∃ x1 .
and
(
x1
∈
omega
)
(
∃ x2 .
and
(
x2
∈
omega
)
(
∃ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
x0
=
add_nat
(
mul_nat
x1
x1
)
(
add_nat
(
mul_nat
x2
x2
)
(
add_nat
(
mul_nat
x3
x3
)
(
mul_nat
x4
x4
)
)
)
)
)
)
)
leaving 2 subgoals.
The subproof is completed by applying H0.
Let x1 of type
ι
be given.
Assume H1:
(
λ x2 .
and
(
x2
∈
omega
)
(
∃ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
∃ x5 .
and
(
x5
∈
omega
)
(
x0
=
add_SNo
(
mul_SNo
x2
x2
)
(
add_SNo
(
mul_SNo
x3
x3
)
(
add_SNo
(
mul_SNo
x4
x4
)
(
mul_SNo
x5
x5
)
)
)
)
)
)
)
)
x1
.
Apply H1 with
∃ x2 .
and
(
x2
∈
omega
)
(
∃ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
∃ x5 .
and
(
x5
∈
omega
)
(
x0
=
add_nat
(
mul_nat
x2
x2
)
(
add_nat
(
mul_nat
x3
x3
)
(
add_nat
(
mul_nat
x4
x4
)
(
mul_nat
x5
x5
)
)
)
)
)
)
)
.
Assume H2:
x1
∈
omega
.
Assume H3:
∃ x2 .
and
(
x2
∈
omega
)
(
∃ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
x0
=
add_SNo
(
mul_SNo
x1
x1
)
(
add_SNo
(
mul_SNo
x2
x2
)
(
add_SNo
(
mul_SNo
x3
x3
)
(
mul_SNo
x4
x4
)
)
)
)
)
)
.
Apply H3 with
∃ x2 .
and
(
x2
∈
omega
)
(
∃ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
∃ x5 .
and
(
x5
∈
omega
)
(
x0
=
add_nat
(
mul_nat
x2
x2
)
(
add_nat
(
mul_nat
x3
x3
)
(
add_nat
(
mul_nat
x4
x4
)
(
mul_nat
x5
x5
)
)
)
)
)
)
)
.
Let x2 of type
ι
be given.
Assume H4:
(
λ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
∃ x5 .
and
(
x5
∈
omega
)
(
x0
=
add_SNo
(
mul_SNo
x1
x1
)
(
add_SNo
(
mul_SNo
x3
x3
)
(
add_SNo
(
mul_SNo
x4
x4
)
(
mul_SNo
x5
x5
)
)
)
)
)
)
)
x2
.
Apply H4 with
∃ x3 .
and
(
x3
∈
omega
)
(
∃ x4 .
and
(
x4
∈
omega
)
(
∃ x5 .
and
(
x5
∈
omega
)
(
∃ x6 .
and
(
x6
∈
omega
)
(
x0
=
add_nat
(
mul_nat
x3
x3
)
(
add_nat
(
mul_nat
x4
x4
)
(
add_nat
(
mul_nat
x5
x5
)
(
mul_nat
x6
x6
)
)
)
)
)
)
)
.
Assume H5:
x2
∈
omega
.
Assume H6:
∃ x3 .
and
(
...
∈
...
)
...
.
...
■